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Let omega(n) be the number of distinct prime factors dividing n and m > nnatural numbers. We calculate a formula showing which prime numbers in whichintervals divide a given binomial coefficient. From this formula we get anidentity omega(binom(nk)(mk))=sum_i (pi(k/b(i))- pi(k/a(i))) + O(sqrt(k)).Erdoes mentioned that omega(binom(nk)(mk))= log n^n/(m^m (n-m)^(n-m)) k/log k +o(k/log k).  As an application of the above identities, we conclude some well-known factsabout the distribution of the primes and deduce for all natural numbers k anexpression (also well-known) log k = sum_i a_k(i) which generalizes log 2 =sum_i^(infty) (-1)^(j+1) / j.

Binomial Coefficients and the Distribution of the Primes